How to estimate the cost and time to complete an AI Project
If you are a freelancer, when a client asks to create a website we can easily measure how much the total cost is needed based on the requirements of the client. (the backend, UI/UX design, features, etc.). We can even measure the estimated time of completion.


This is only the case when the full scope and design of the site is a well understood and relatively standard thing. With more experience, and perhaps a dedicated team, more complex sites and features can be addressed like this. It is just as common in my experience to view ongoing development of a site as a series of related projects, each of which can be estimated and costed as they become more feasible and closer to starting. 

I have never seen a stand-alone site fully costed and estimated in my own work, because I work in-house on more complex features than this is possible for, but am aware this is a common approach, especially for agencies and freelancers. The difference is that the people involved are experts on producing well-defined work, and that in order to make a pre-estimated or even fixed price sale, they have put a lot of effort into risk management. Some of the risk management is contractual (e.g. "maximum of 3 major revisions" for web site design), some of it is in requirements gathering - often an initial pre-sales consultancy meeting is arranged, and/or a project proposal document generated where a lot of the possible misunderstandings between customer and developer are dealt with, and it is made clear what expectations are on both sides.

For a freelancer or agency, this de-risking is also spread over multiple projects as they will develop solutions that can be re-used including standardised contracts and templates for project planning etc.

There are also a few different project management approaches that deal with initial uncertainty in different ways. A "waterfall" approach of pre-planning as much in advance as possible requires very good knowledge of the work that will occur in later stages of a project, which of course depends heavily on earlier stages meeting all expectations of both the customer and developer. An "agile" approach will acknowledge that there are unknowns, and attempt to address them with some form of just in time planning. The problem with an agile approach is that "just in time" may be far too late if you need to budget up front (although it is still possible for instance by reducing scope to fit, provided that is agreed). 


  What if the client asks you to make an AI project (image recognition, speech recognition, or NLP), how do you tell the client the estimated cost and time needed to complete the project in the beginning?


There is no difference in this respect just because it is an AI project. This would apply to any software project, including web development, where one or more of the following is true:


Scope of work is not clear
There are technical hurdles with unknown resolutions. I think it is likely you are focusing on this for your imagined AI project.
Customer expectations are not clear
Project success criteria is not measurable, or not known if achievable


As any software developer, if you are faced with project work with these features, you do some pre-project tasks focused on de-risking the project before attempting it. There are a few different approaches, than can be used in combination:


Gather more requirements, spend more time up front consulting on the project
Test assumptions before starting project. E.g. if you are not sure about quality of the data, take a look at it as early as possible. If you don't know if technology X can do Y, investigate it in advance.
Work on a timesheet basis, not fixed cost. Alternatively fix cost (or have a maximum), but leave scope of work open.
Have a flexible project plan, and re-estimate remaining work routinely.
Put any assumptions about customer support for the project, such as availablilty and quality of data, into project plan and in general form into your standard contract (e.g. something legal along the lines of "Project delivery depends on customer providing resources as outlined in the project plan document. If the resources are not provided in reasonable time, then the project may over-run or have additional costs, which will be covered by the customer, and not by the freelancer")


For additional up-front work on project feasibilty and scope, if you are concerned about risk to yourself as a freelancer (because customer expects these as a pre-sale effort), you could propose to bill for that as consultancy work, prior to starting project proper. You would likely need to produce something like a project proposal document in order that the paying customer received something for spending this money. Producing such a document might even show both you and the customer that the project is not feasible without additional work - in whch case you could note the steps needed to make it feasible (e.g. customer needs to source 10,000 more training images without the flaws noticed in existing images).